Mathematical Methods for Physicists: A concise introduction - Site Map

HERMITE'S EQUATION
Rodrigues' formula for Hermite polynomials H n …x†
The Hermite polynomials are also given by the formula
H n …x† ˆ…1† n e x2
dn
dx n …ex2 †:
To prove this formula, let us write q ˆ e x2 . Then
Dq ‡ 2xq ˆ 0;
D ˆ d
dx :
Di€erentiate this (n ‡ 1) times by the Leibnitz' rule giving
…7:42†
Writing y ˆ…1† n D n q gives
D n‡2 q ‡ 2xD n‡1 q ‡ 2…n ‡ 1†D n q ˆ 0:
D 2 y ‡ 2xDy ‡ 2…n ‡ 1†y ˆ 0
…7:43†
substitute u ˆ e x2 y then
Du ˆ e x2 f2xy ‡ Dyg
and
D 2 u ˆ e x2 fD 2 y ‡ 4xDy ‡ 4x 2 y ‡ 2yg:
Hence by Eq. (7.43) we get
D 2 u 2xDu ‡ 2nu ˆ 0;
which indicates that
u ˆ…1† n e x2 D n …e x2 †
is a polynomial solution of Hermite's equation (7.38).
Recurrence relations for Hermite polynomials
Rodrigues' formula gives on di€erentiation
that is,
H 0
n…x† ˆ…1† n 2xe x2 D n …e x2 †‡…1† n e x2 D n‡1 …e x2 †:
Eq. (7.44) gives on di€erentiation
H 0
n…x† ˆ2xH n …x†H n‡1 …x†:
H 00
n …x† ˆ2H n …x†‡2xH 0
n…x†H 0
n‡1…x†:
…7:44†
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